X:set of points in the plane,no 3 on a line Va,b,c∈X,△abc:points in triangle abc f({a,b,c})=△abe mod2 2 X=R(2;m,m)f:(3)→{0,1}3SCX,|S1=m S∈(X) monochromatic(③) S is a convex m-gon Otherwise,b disjoint union: Contradiction! △abe=△abdU△acdU△bcdU{d} f(abc)=f(abd)+f(acd)+f(bcd)+1 C|X| = R3(2; m, m) ￾S 3 ⇥ monochromatic ⇥f : ￾X 3 ⇥ ￾ {0, 1} ⇥S ￾ ￾X m ⇥ X: set of points in the plane, no 3 on a line f({a, b, c}) = |￾abc| mod 2 ⇥a, b, c ￾ X, ￾abc: points in triangle abc a b c S is a convex m-gon Otherwise, a b c d ￾abc = ￾abd ⇥ ￾acd ⇥ ￾bcd disjoint union: f(abc) = f(abd) + f(acd) + f(bcd)+1 ￾{d} Contradiction! ￾S ￾ X, |S| = m